Consensus

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About the power to enforce and prevent Consensus - by ... - Janlo.de

Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsAbout the power to enforce and preventConsensusby manipulation of the communication regime in theWeisbuch-Deffuant model of continuous opinion dynamicsunder bounded confidenceJan Lorenz 1 Diemo Urbig 21 Department of Mathematics and Computer Science, Universität BremenScholarship of Foundation Friederich Ebert, Bonn2 Department of Computer Science and School of Business and EconomicsHumboldt Humboldt-Universität zu BerlinDPG AKSOE, Dresden 27-03-2006Jan Lorenz, Diemo Urbig Enforcing Consensus


OutlineContinuous opinion dynamicsMathematicsAgent-based rules for communicationImplications1 Continuous opinion dynamics2 Mathematics3 Agent-based rules for communication4 ImplicationsJan Lorenz, Diemo UrbigEnforcing Consensus


OutlineContinuous opinion dynamicsMathematicsAgent-based rules for communicationImplications1 Continuous opinion dynamics2 Mathematics3 Agent-based rules for communication4 ImplicationsJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsContinuous opinion dynamics, gossip communicationThe Weisbuch-Deffuant modelDefinition (Weisbuch-Deffuant Model)Given x(0) ∈ R n , ε > 0 bound of confidence,0 < µ ≤ 0.5 cautiousness we defineWeisbuch-Deffuant process of opinion dynamicsas the random process (x(t)) t∈N0 that chooses ineach time step two random agents i, j whichperformif |x i (t) − x j (t)| < εx i (t + 1) = (1 − µ)x i (t) + µx j (t),x j (t + 1) = µx i (t) + (1 − µ)x j (t)Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsContinuous opinion dynamics, gossip communicationThe Weisbuch-Deffuant modelDefinition (Weisbuch-Deffuant Model)Given x(0) ∈ R n , ε > 0 bound of confidence,0 < µ ≤ 0.5 cautiousness we defineWeisbuch-Deffuant process of opinion dynamicsas the random process (x(t)) t∈N0 that chooses ineach time step two random agents i, j whichperformif |x i (t) − x j (t)| < εx i (t + 1) = (1 − µ)x i (t) + µx j (t),x j (t + 1) = µx i (t) + (1 − µ)x j (t)Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsContinuous opinion dynamics, gossip communicationThe Weisbuch-Deffuant modelDefinition (Weisbuch-Deffuant Model)Given x(0) ∈ R n , ε > 0 bound of confidence,0 < µ ≤ 0.5 cautiousness we defineWeisbuch-Deffuant process of opinion dynamicsas the random process (x(t)) t∈N0 that chooses ineach time step two random agents i, j whichperformif |x i (t) − x j (t)| < εx i (t + 1) = (1 − µ)x i (t) + µx j (t),x j (t + 1) = µx i (t) + (1 − µ)x j (t)Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsContinuous opinion dynamics, gossip communicationThe Weisbuch-Deffuant modelDefinition (Weisbuch-Deffuant Model)Given x(0) ∈ R n , ε > 0 bound of confidence,0 < µ ≤ 0.5 cautiousness we defineWeisbuch-Deffuant process of opinion dynamicsas the random process (x(t)) t∈N0 that chooses ineach time step two random agents i, j whichperformif |x i (t) − x j (t)| < εx i (t + 1) = (1 − µ)x i (t) + µx j (t),x j (t + 1) = µx i (t) + (1 − µ)x j (t)Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsContinuous opinion dynamics, gossip communicationThe Weisbuch-Deffuant modelDefinition (Weisbuch-Deffuant Model)Given x(0) ∈ R n , ε > 0 bound of confidence,0 < µ ≤ 0.5 cautiousness we defineWeisbuch-Deffuant process of opinion dynamicsas the random process (x(t)) t∈N0 that chooses ineach time step two random agents i, j whichperformif |x i (t) − x j (t)| < εx i (t + 1) = (1 − µ)x i (t) + µx j (t),x j (t + 1) = µx i (t) + (1 − µ)x j (t)Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsContinuous opinion dynamics, gossip communicationThe Weisbuch-Deffuant modelDefinition (Weisbuch-Deffuant Model)Given x(0) ∈ R n , ε > 0 bound of confidence,0 < µ ≤ 0.5 cautiousness we defineWeisbuch-Deffuant process of opinion dynamicsas the random process (x(t)) t∈N0 that chooses ineach time step two random agents i, j whichperformif |x i (t) − x j (t)| < εx i (t + 1) = (1 − µ)x i (t) + µx j (t),x j (t + 1) = µx i (t) + (1 − µ)x j (t)Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsContinuous opinion dynamics, gossip communicationThe Weisbuch-Deffuant modelDefinition (Weisbuch-Deffuant Model)Given x(0) ∈ R n , ε > 0 bound of confidence,0 < µ ≤ 0.5 cautiousness we defineWeisbuch-Deffuant process of opinion dynamicsas the random process (x(t)) t∈N0 that chooses ineach time step two random agents i, j whichperformif |x i (t) − x j (t)| < εx i (t + 1) = (1 − µ)x i (t) + µx j (t),x j (t + 1) = µx i (t) + (1 − µ)x j (t)Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsContinuous opinion dynamics, gossip communicationThe Weisbuch-Deffuant modelDefinition (Weisbuch-Deffuant Model)Given x(0) ∈ R n , ε > 0 bound of confidence,0 < µ ≤ 0.5 cautiousness we defineWeisbuch-Deffuant process of opinion dynamicsas the random process (x(t)) t∈N0 that chooses ineach time step two random agents i, j whichperformif |x i (t) − x j (t)| < εx i (t + 1) = (1 − µ)x i (t) + µx j (t),x j (t + 1) = µx i (t) + (1 − µ)x j (t)Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsGeneral features: Clusters and minoritiesExample and Bifurcation diagramexample: n = 200, ε = 0.16, µ = 0.5bifurcation diagram of interactive Markov chain1100+ε0.880−εWD model0.60.4+ε−ε60400.220+ε01000 2000 3000 4000 5000t00.05 0.1 0.15 0.2 0.25 0.3εJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBut the communication regime can matter!Examples without random communication: Normal random communication1n = 100, ε = 0.16, µ = 0.50.80.60.40.201 3000Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBut the communication regime can matter!Examples without random communication: Produce more clusters1n = 100, ε = 0.16, µ = 0.50.80.60.40.201 2000more clusters possible!Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBut the communication regime can matter!Examples without random communication: Enforce Consensus1n = 100, ε = 0.16, µ = 0.50.80.60.40.201 100000also possible for ε = 0.047!Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsSo, let’s start an investigationOur questionsTo what extend can the communication regime manipulatethe outcome?For what ε can we enforce (or prevent) consensus?Are there agent-based rules for communication regimesthat foster or weaken the chances for consensus?Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsSo, let’s start an investigationOur questionsTo what extend can the communication regime manipulatethe outcome?For what ε can we enforce (or prevent) consensus?Are there agent-based rules for communication regimesthat foster or weaken the chances for consensus?Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsSo, let’s start an investigationOur questionsTo what extend can the communication regime manipulatethe outcome?For what ε can we enforce (or prevent) consensus?Are there agent-based rules for communication regimesthat foster or weaken the chances for consensus?Jan Lorenz, Diemo UrbigEnforcing Consensus


OutlineContinuous opinion dynamicsMathematicsAgent-based rules for communicationImplications1 Continuous opinion dynamics2 Mathematics3 Agent-based rules for communication4 ImplicationsJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsε low and ε highDefinition and results about ε low and ε highGiven an initial opinion profile x(0) defineε low The lowest ε such that consensus is possible withat least one communication regime.max ∆x(0) ≤ ε low ≤1 − µ⌈range(x(0))max ∆x(0) ⌉1 − µmax ∆x(0)ε high The highest ε such that a stable dissent is possiblewith at least one communication regime.⎛⎞ε high = max ⎝ 1 n∑x i (0) − 1 k∑x j (0) ⎠k∈n−1 n − kki=k+1j=1Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsε low and ε highDefinition and results about ε low and ε highGiven an initial opinion profile x(0) defineε low The lowest ε such that consensus is possible withat least one communication regime.max ∆x(0) ≤ ε low ≤1 − µ⌈range(x(0))max ∆x(0) ⌉1 − µmax ∆x(0)ε high The highest ε such that a stable dissent is possiblewith at least one communication regime.⎛⎞ε high = max ⎝ 1 n∑x i (0) − 1 k∑x j (0) ⎠k∈n−1 n − kki=k+1j=1Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsε low and ε highDefinition and results about ε low and ε highGiven an initial opinion profile x(0) defineε low The lowest ε such that consensus is possible withat least one communication regime.max ∆x(0) ≤ ε low ≤1 − µ⌈range(x(0))max ∆x(0) ⌉1 − µmax ∆x(0)ε high The highest ε such that a stable dissent is possiblewith at least one communication regime.⎛⎞ε high = max ⎝ 1 n∑x i (0) − 1 k∑x j (0) ⎠k∈n−1 n − kki=k+1j=1Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsε low and ε highDefinition and results about ε low and ε highGiven an initial opinion profile x(0) defineε low The lowest ε such that consensus is possible withat least one communication regime.max ∆x(0) ≤ ε low ≤1 − µ⌈range(x(0))max ∆x(0) ⌉1 − µmax ∆x(0)ε high The highest ε such that a stable dissent is possiblewith at least one communication regime.⎛⎞ε high = max ⎝ 1 n∑x i (0) − 1 k∑x j (0) ⎠k∈n−1 n − kki=k+1j=1Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsε low and ε highDefinition and results about ε low and ε highGiven an initial opinion profile x(0) defineε low The lowest ε such that consensus is possible withat least one communication regime.max ∆x(0) ≤ ε low ≤1 − µ⌈range(x(0))max ∆x(0) ⌉1 − µmax ∆x(0)ε high The highest ε such that a stable dissent is possiblewith at least one communication regime.⎛⎞ε high = max ⎝ 1 n∑x i (0) − 1 k∑x j (0) ⎠k∈n−1 n − kki=k+1j=1Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplications’Everything’ is possible by manipulation of thecommunication regime!Relevance of ε low and ε highQuick check: Take 500 random initial profilesCompute bounds on ε low and ε high .Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplications’Everything’ is possible by manipulation of thecommunication regime!Relevance of ε low and ε highQuick check: Take 500 random initial profilesCompute bounds on ε low and ε high .Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplications’Everything’ is possible by manipulation of thecommunication regime!Relevance of ε low and ε highQuick check: Take 500 random initial profilesCompute bounds on ε low and ε high .Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplications’Everything’ is possible by manipulation of thecommunication regime!Relevance of ε low and ε highQuick check: Take 500 random initial profilesCompute bounds on ε low and ε high .ε−phases1pluralism2 clust.1 cluster = consensus231 and 2 clusters possible, n = 10001 and 2 clusters possible, n = 2000 0.1 0.2 0.3 0.4 0.5 0.6Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsFortunato’s universality of consensus threshold issimply provedConjecture (Fortunato)For ε > 0.5 we reach consensus regardless of network. 1Take the opinion spaceDivide it under ’maximisation’Lemma: The mean conserves!For uniform distribution in the limit of large n⇒ ε high = 0.51 Fortunato, Santo; Int. J. of Modern Physics C, 2004, 15, 1301-1307Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsFortunato’s universality of consensus threshold issimply provedConjecture (Fortunato)For ε > 0.5 we reach consensus regardless of network. 1Take the opinion spaceDivide it under ’maximisation’Lemma: The mean conserves!For uniform distribution in the limit of large n⇒ ε high = 0.51 Fortunato, Santo; Int. J. of Modern Physics C, 2004, 15, 1301-1307Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsFortunato’s universality of consensus threshold issimply provedConjecture (Fortunato)For ε > 0.5 we reach consensus regardless of network. 1Take the opinion spaceDivide it under ’maximisation’Lemma: The mean conserves!For uniform distribution in the limit of large n⇒ ε high = 0.51 Fortunato, Santo; Int. J. of Modern Physics C, 2004, 15, 1301-1307Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsFortunato’s universality of consensus threshold issimply provedConjecture (Fortunato)For ε > 0.5 we reach consensus regardless of network. 1Take the opinion spaceDivide it under ’maximisation’Lemma: The mean conserves!For uniform distribution in the limit of large n⇒ ε high = 0.51 Fortunato, Santo; Int. J. of Modern Physics C, 2004, 15, 1301-1307Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsFortunato’s universality of consensus threshold issimply provedConjecture (Fortunato)For ε > 0.5 we reach consensus regardless of network. 1Take the opinion spaceDivide it under ’maximisation’Lemma: The mean conserves!For uniform distribution in the limit of large n⇒ ε high = 0.51 Fortunato, Santo; Int. J. of Modern Physics C, 2004, 15, 1301-1307Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsFortunato’s universality of consensus threshold issimply provedConjecture (Fortunato)For ε > 0.5 we reach consensus regardless of network. 1Take the opinion spaceDivide it under ’maximisation’Lemma: The mean conserves!For uniform distribution in the limit of large n⇒ ε high = 0.51 Fortunato, Santo; Int. J. of Modern Physics C, 2004, 15, 1301-1307Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsFortunato’s universality of consensus threshold issimply provedConjecture (Fortunato)For ε > 0.5 we reach consensus regardless of network. 1Take the opinion spaceDivide it under ’maximisation’Lemma: The mean conserves!For uniform distribution in the limit of large n⇒ ε high = 0.51 Fortunato, Santo; Int. J. of Modern Physics C, 2004, 15, 1301-1307Jan Lorenz, Diemo UrbigEnforcing Consensus


OutlineContinuous opinion dynamicsMathematicsAgent-based rules for communicationImplications1 Continuous opinion dynamics2 Mathematics3 Agent-based rules for communication4 ImplicationsJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCommunication Chain, Balancing and Curious AgentsDefinition of agent-based rulesCommunication Chain One agent starts. After successfulcompromise, the other goes on in the otherdirection.Balancing Agents After successful compromise, try to findsomeone from the other side.Curious Agents After successful compromise, try to findsomeone from the same side.General After f max (frustration maximum) unsuccessfultries, neglect the rule.Interesting For µ = 0.5 balancing ’=’ curiousJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCommunication Chain, Balancing and Curious AgentsDefinition of agent-based rulesCommunication Chain One agent starts. After successfulcompromise, the other goes on in the otherdirection.Balancing Agents After successful compromise, try to findsomeone from the other side.Curious Agents After successful compromise, try to findsomeone from the same side.General After f max (frustration maximum) unsuccessfultries, neglect the rule.Interesting For µ = 0.5 balancing ’=’ curiousJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCommunication Chain, Balancing and Curious AgentsDefinition of agent-based rulesCommunication Chain One agent starts. After successfulcompromise, the other goes on in the otherdirection.Balancing Agents After successful compromise, try to findsomeone from the other side.Curious Agents After successful compromise, try to findsomeone from the same side.General After f max (frustration maximum) unsuccessfultries, neglect the rule.Interesting For µ = 0.5 balancing ’=’ curiousJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCommunication Chain, Balancing and Curious AgentsDefinition of agent-based rulesCommunication Chain One agent starts. After successfulcompromise, the other goes on in the otherdirection.Balancing Agents After successful compromise, try to findsomeone from the other side.Curious Agents After successful compromise, try to findsomeone from the same side.General After f max (frustration maximum) unsuccessfultries, neglect the rule.Interesting For µ = 0.5 balancing ’=’ curiousJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCommunication Chain, Balancing and Curious AgentsDefinition of agent-based rulesCommunication Chain One agent starts. After successfulcompromise, the other goes on in the otherdirection.Balancing Agents After successful compromise, try to findsomeone from the other side.Curious Agents After successful compromise, try to findsomeone from the same side.General After f max (frustration maximum) unsuccessfultries, neglect the rule.Interesting For µ = 0.5 balancing ’=’ curiousJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCommunication Chain, Balancing and Curious AgentsDefinition of agent-based rulesCommunication Chain One agent starts. After successfulcompromise, the other goes on in the otherdirection.Balancing Agents After successful compromise, try to findsomeone from the other side.Curious Agents After successful compromise, try to findsomeone from the same side.General After f max (frustration maximum) unsuccessfultries, neglect the rule.Interesting For µ = 0.5 balancing ’=’ curiousJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsSimulation SetupFor Communication Chain, Balancing and Curious AgentsParameter spacen = 200µ = 0.2, 0.5ε = 0, +0.01 . . . , 0.35f max = 0, 1, 2, 4, 8, 16, 323000 independent simulation runs for each point in theparameter space, with random initial profiles and randomcommunication partners.Observe: Size of biggest cluster after stabilization.Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsThe Communication ChainSimulation Results µ = 0.5, f max = 0200chain, µ = 0.5average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε0Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsThe Communication ChainSimulation Results µ = 0.5, f max = 0, 1200chain, µ = 0.5average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε01Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsThe Communication ChainSimulation Results µ = 0.5, f max = 0, 1, 2200chain, µ = 0.5average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε012Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsThe Communication ChainSimulation Results µ = 0.5, f max = 0, 1, 2, 4200chain, µ = 0.5average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε0124Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsThe Communication ChainSimulation Results µ = 0.5, f max = 0, 1, 2, 4, 8200chain, µ = 0.5average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε01248Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsThe Communication ChainSimulation Results µ = 0.5, f max = 0, 1, 2, 4, 8, 16200chain, µ = 0.5average size of biggest cluster15010001502481600 0.05 0.1 0.15 0.2 0.25 0.3εJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsThe Communication ChainSimulation Results µ = 0.5, f max = 0, 1, 2, 4, 8, 16, 32200chain, µ = 0.5average size of biggest cluster1501000125048163200 0.05 0.1 0.15 0.2 0.25 0.3εJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing/Curious AgentsSimulation Results µ = 0.5, f max = 0200balancing/curious, µ = 0.5average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε0Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing/Curious AgentsSimulation Results µ = 0.5, f max = 0, 1200balancing/curious, µ = 0.5average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε01Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing/Curious AgentsSimulation Results µ = 0.5, f max = 0, 1, 2200balancing/curious, µ = 0.5average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε012Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing/Curious AgentsSimulation Results µ = 0.5, f max = 0, 1, 2, 4200balancing/curious, µ = 0.5average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε0124Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing/Curious AgentsSimulation Results µ = 0.5, f max = 0, 1, 2, 4, 8200balancing/curious, µ = 0.5average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε01248Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing/Curious AgentsSimulation Results µ = 0.5, f max = 0, 1, 2, 4, 8, 16200balancing/curious, µ = 0.5average size of biggest cluster15010001502481600 0.05 0.1 0.15 0.2 0.25 0.3εJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing/Curious AgentsSimulation Results µ = 0.5, f max = 0, 1, 2, 4, 8, 16, 32200balancing/curious, µ = 0.5average size of biggest cluster1501000125048163200 0.05 0.1 0.15 0.2 0.25 0.3εJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing Agents more cautiousSimulation Results µ = 0.2, f max = 0200balancing, µ = 0.2average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε0Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1200balancing, µ = 0.2average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε01Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1, 2200balancing, µ = 0.2average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε012Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1, 2, 4200balancing, µ = 0.2average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε0124Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8200balancing, µ = 0.2average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε01248Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8, 16200balancing, µ = 0.2average size of biggest cluster15010001502481600 0.05 0.1 0.15 0.2 0.25 0.3εJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsBalancing Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8, 16, 32200balancing, µ = 0.2average size of biggest cluster1501000125048163200 0.05 0.1 0.15 0.2 0.25 0.3εJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCurious Agents more cautiousSimulation Results µ = 0.2, f max = 0200curious, µ = 0.2average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε0Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCurious Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1200curious, µ = 0.2average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε01Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCurious Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1, 2200curious, µ = 0.2average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε012Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCurious Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1, 2, 4200curious, µ = 0.2average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε0124Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCurious Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8200curious, µ = 0.2average size of biggest cluster1501005000 0.05 0.1 0.15 0.2 0.25 0.3ε01248Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCurious Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8, 16200curious, µ = 0.2average size of biggest cluster15010001502481600 0.05 0.1 0.15 0.2 0.25 0.3εJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCurious Agents more cautiousSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8, 16, 32200curious, µ = 0.2average size of biggest cluster1501000125048163200 0.05 0.1 0.15 0.2 0.25 0.3εJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCautious Balancing Agents, ExamplesSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8, 16, 32ε = 0.25, µ = 0.2, f max= 010.90.80.70.60.50.4+ε0.30.20.10−ε5000 10000Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCautious Balancing Agents, ExamplesSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8, 16, 32ε = 0.25, µ = 0.2, balancing, f max= 410.90.80.70.60.50.40.3+ε−ε0.20.1010000 20000Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCautious Curious Agents, ExamplesSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8, 16, 32ε = 0.3, µ = 0.2, f max= 010.90.80.70.6+ε0.50.40.3−ε0.20.105000Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCautious Curious Agents, ExamplesSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8, 16, 3210.9ε = 0.3, µ = 0.2, curious, f max= 40.80.7+ε0.60.50.4−ε0.30.20.105000 10000Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsCautious Curious Agents, ExamplesSimulation Results µ = 0.2, f max = 0, 1, 2, 4, 8, 16, 32ε = 0.25, µ = 0.2, curious, f max= 1610.90.80.70.6+ε0.50.40.3−ε0.20.1015000 30000Jan Lorenz, Diemo UrbigEnforcing Consensus


OutlineContinuous opinion dynamicsMathematicsAgent-based rules for communicationImplications1 Continuous opinion dynamics2 Mathematics3 Agent-based rules for communication4 ImplicationsJan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsImplications of the resultsSummaryThe number of clusters can be manipulated to a very largeextend with hand-picked communication regimes.But agent-based rules can also foster consensus very well.They prevent early clustering and produce a ’slower ismore consensual’-effect.If you want your agents to foster consensus by beingBalancing, appeal to them to be cautious.If you want your agents to foster consensus by beingCurious, appeal to them to be not cautious.Otherwise, you may get a negative effect under lowfrustration tolerance.The impact of Balancing is higher than for Curious.Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsImplications of the resultsSummaryThe number of clusters can be manipulated to a very largeextend with hand-picked communication regimes.But agent-based rules can also foster consensus very well.They prevent early clustering and produce a ’slower ismore consensual’-effect.If you want your agents to foster consensus by beingBalancing, appeal to them to be cautious.If you want your agents to foster consensus by beingCurious, appeal to them to be not cautious.Otherwise, you may get a negative effect under lowfrustration tolerance.The impact of Balancing is higher than for Curious.Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsImplications of the resultsSummaryThe number of clusters can be manipulated to a very largeextend with hand-picked communication regimes.But agent-based rules can also foster consensus very well.They prevent early clustering and produce a ’slower ismore consensual’-effect.If you want your agents to foster consensus by beingBalancing, appeal to them to be cautious.If you want your agents to foster consensus by beingCurious, appeal to them to be not cautious.Otherwise, you may get a negative effect under lowfrustration tolerance.The impact of Balancing is higher than for Curious.Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsImplications of the resultsSummaryThe number of clusters can be manipulated to a very largeextend with hand-picked communication regimes.But agent-based rules can also foster consensus very well.They prevent early clustering and produce a ’slower ismore consensual’-effect.If you want your agents to foster consensus by beingBalancing, appeal to them to be cautious.If you want your agents to foster consensus by beingCurious, appeal to them to be not cautious.Otherwise, you may get a negative effect under lowfrustration tolerance.The impact of Balancing is higher than for Curious.Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsImplications of the resultsSummaryThe number of clusters can be manipulated to a very largeextend with hand-picked communication regimes.But agent-based rules can also foster consensus very well.They prevent early clustering and produce a ’slower ismore consensual’-effect.If you want your agents to foster consensus by beingBalancing, appeal to them to be cautious.If you want your agents to foster consensus by beingCurious, appeal to them to be not cautious.Otherwise, you may get a negative effect under lowfrustration tolerance.The impact of Balancing is higher than for Curious.Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsImplications of the resultsSummaryThe number of clusters can be manipulated to a very largeextend with hand-picked communication regimes.But agent-based rules can also foster consensus very well.They prevent early clustering and produce a ’slower ismore consensual’-effect.If you want your agents to foster consensus by beingBalancing, appeal to them to be cautious.If you want your agents to foster consensus by beingCurious, appeal to them to be not cautious.Otherwise, you may get a negative effect under lowfrustration tolerance.The impact of Balancing is higher than for Curious.Jan Lorenz, Diemo UrbigEnforcing Consensus


Continuous opinion dynamicsMathematicsAgent-based rules for communicationImplicationsImplications of the resultsSummaryThe number of clusters can be manipulated to a very largeextend with hand-picked communication regimes.But agent-based rules can also foster consensus very well.They prevent early clustering and produce a ’slower ismore consensual’-effect.If you want your agents to foster consensus by beingBalancing, appeal to them to be cautious.If you want your agents to foster consensus by beingCurious, appeal to them to be not cautious.Otherwise, you may get a negative effect under lowfrustration tolerance.The impact of Balancing is higher than for Curious.Jan Lorenz, Diemo UrbigEnforcing Consensus


AppendixReferencesModel reference:G. Deffuant, J. P. Nadal, F. Amblard, and G. Weisbuch.Mixing beliefs among interacting agents.Advances in Complex Systems, 3:87–98, 2000.More papers on continuous opinion dynamics, joint andindividual work:Jan Lorenz.www.janlo.deThank you for your attention!Diemo Urbig.www.diemo.deJan Lorenz, Diemo UrbigEnforcing Consensus


AppendixReferencesModel reference:G. Deffuant, J. P. Nadal, F. Amblard, and G. Weisbuch.Mixing beliefs among interacting agents.Advances in Complex Systems, 3:87–98, 2000.More papers on continuous opinion dynamics, joint andindividual work:Jan Lorenz.www.janlo.deThank you for your attention!Diemo Urbig.www.diemo.deJan Lorenz, Diemo UrbigEnforcing Consensus

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